Consider the plane in R3 given by the equation x-2y-z=0. (a) (2 points) Obtain two vectors...

   
   

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Consider the plane in R3 given by the equation x-2y-z=0. (a) (2 points) Obtain two vectors P1 and p2 that form a basis for this plane. These vectors need not be orthonormal. (3 points) Using your answer to (a), obtain vectors q and q2 that form an orthonormal basis for this plane. q1= 92 = 3. (5 points total) Consider the set of all vectors x = (x, y, z, w) = R4 that lie on the hyperplane x-w=0. (a) (2 points) Obtain three vectors P1, P2, and p3 that form a basis for this hyperplane. These vectors need not be orthonormal. (Be sure to check your vectors satisfy the definition of a basis.) P1 --0-0-0 P3 = (b) (3 points) Using your answer from (a), obtain vectors 91, 92, and 3 that form an orthonormal basis for this hyperplane. --0--0--0 2. (5 points total) Consider the following matrix: A = (39) 4 (a) (4 points) Obtain the QR factorization of A. Show each step of Gram-Schmidt! 0 R = (b) (1 point) Use your QR factorization to solve the system Ax = b, where b = (5/5). 5. (5 points total) The temperature on a chilly Seattle day is modeled by T(t) = a sin(t) + b, where the coefficients a and b are to be determined and t is time. Suppose we have the following data: t T 0 1 1/6 -1/2 1/2 0 (a) (1 point) Based on the data above, construct a 3 2 linear system for a and b. (b) (4 points) Find the least-squares solution of your system in (a). (5 points total) Consider the following matrix: 2 1 A = 1 2 0 2-29 (a) (4 points) Obtain the reduced QR factorization of A. Show each step of Gram-Schmidt! (b) (1 point) Based on this QR factorization, is A invertible? Why or why not? This should be a one sentence answer. 5. (5 points total) Suppose we collect the following data: I y 0 1 1 2 2 0 We wish to fit to this data the linear polynomial y = ax + b. (a) (1 point) Determine a linear system for the coefficients a and b. Consider the plane in R3 given by the equation x-2y-z=0. (a) (2 points) Obtain two vectors P1 and p2 that form a basis for this plane. These vectors need not be orthonormal. (3 points) Using your answer to (a), obtain vectors q and q2 that form an orthonormal basis for this plane. q1= 92 = 3. (5 points total) Consider the set of all vectors x = (x, y, z, w) = R4 that lie on the hyperplane x-w=0. (a) (2 points) Obtain three vectors P1, P2, and p3 that form a basis for this hyperplane. These vectors need not be orthonormal. (Be sure to check your vectors satisfy the definition of a basis.) P1 --0-0-0 P3 = (b) (3 points) Using your answer from (a), obtain vectors 91, 92, and 3 that form an orthonormal basis for this hyperplane. --0--0--0 2. (5 points total) Consider the following matrix: A = (39) 4 (a) (4 points) Obtain the QR factorization of A. Show each step of Gram-Schmidt! 0 R = (b) (1 point) Use your QR factorization to solve the system Ax = b, where b = (5/5). 5. (5 points total) The temperature on a chilly Seattle day is modeled by T(t) = a sin(t) + b, where the coefficients a and b are to be determined and t is time. Suppose we have the following data: t T 0 1 1/6 -1/2 1/2 0 (a) (1 point) Based on the data above, construct a 3 2 linear system for a and b. (b) (4 points) Find the least-squares solution of your system in (a). (5 points total) Consider the following matrix: 2 1 A = 1 2 0 2-29 (a) (4 points) Obtain the reduced QR factorization of A. Show each step of Gram-Schmidt! (b) (1 point) Based on this QR factorization, is A invertible? Why or why not? This should be a one sentence answer. 5. (5 points total) Suppose we collect the following data: I y 0 1 1 2 2 0 We wish to fit to this data the linear polynomial y = ax + b. (a) (1 point) Determine a linear system for the coefficients a and b.